2.03 Domain and range

We have been exploring how functions and relations can be used to describe ways of linking two sets of data. We also learnt that relations can only be functions if there's only one $y$y value associated with each $x$x value.

For example, $\left\{\left(1,2\right),\left(5,3\right),\left(2,-7\right),\left(5,-1\right)\right\}${(1,2),(5,3),(2,−7),(5,−1)} is not a function as the same $x$x value of $5$5 can be related to the $y$y values of both $3$3 and $-1$−1. We also know that graphically, we can use the vertical line test to see if a relation is a function.

Domain and range give us a way to describe the set of inputs and outputs for relations. These two words sound similar, don't they? They both seem to be talking about areas and spans, and in maths they have similar definitions:

There are a number of ways to find the domain and range of a relation. One is to look at the coordinates given and simply list the possible values. For example in the previous example, the domain is $\left\{1,5,2\right\}${1,5,2} and the range is $\left\{2,3,-7,-1\right\}${2,3,−7,−1}. Notice how repeated values are not included and order is not important, as we only care about the POSSIBLE values of $x$x and $y$y. 

The other method is to look at a relation graphically, and see how 'wide' or 'long' the graph is:

Horizontally this graph spans from $-1$−1 to $1$1, so we can write the domain as $-1\le x\le1$−1≤x≤1. Similarly, the graph goes vertically from $-2$−2 to $2$2 so the range can be written as $-2\le y\le2$−2≤y≤2.

 

Practice questions

Question 1

question 2

 

Natural domain of an equation

A relation or function may be given as an equation, for example $f(x)=x^2+1$f(x)=x2+1. The equation may have implied restrictions because of the type of equation or restrictions implied or defined for the context of the problem.

Firstly, when looking for the implied or 'natural domain' of an equation we can ask "Is there any input value the equation cannot handle?".

Exploration

The function $f\left(x\right)=\sqrt{x}$f(x)=√x, defined for real numbers, cannot handle negative real numbers. Hence, its domain is: $x\ge0$x≥0.

The function $f\left(x\right)=\frac{1}{x}$f(x)=1x​ cannot handle zero. Hence, its domain is the compound interval: $\left(-\infty,0\right)\cup\left(0,\infty\right)$(−∞,0)∪(0,∞)

Our first example, $f\left(x\right)=x^2+1$f(x)=x2+1 has no exclusions and so its natural domain includes all real numbers: $\left(-\infty,\infty\right)$(−∞,∞).

Secondly, there might be what could be called user-defined restrictions where certain restrictions are put onto the domain so that the function makes sense in the real world.

For example, we might decide to define our function $f\left(x\right)=3.5x$f(x)=3.5x with a domain given by the set of positive integers, because we are actually using the function to determine the revenue on selling $x$x apples. Again we might define the function $f\left(t\right)=200t-4.9t^2$f(t)=200t−4.9t2 where $t$t is the elapsed time in seconds after the stroke of midnight on 31 December 2015. Negative time makes no sense, so we restrict the input to the time domain  $t\ge0$t≥0. 

Worked example

Find the natural domain of $f\left(x\right)=\sqrt{x-x^2}$f(x)=√x−x2, defined over the reals. 

If we try $x=2$x=2, we immediately find a problem, since $f\left(2\right)=\sqrt{2-2^2}=\sqrt{-2}$f(2)=√2−22=√−2. Clearly the natural domain is some subset of the real numbers that doesn't include $2$2. But how do we find it?

We know that we need $x-x^2\ge0$x−x2≥0, or after factorising, $x\left(1-x\right)\ge0$x(1−x)≥0. In words this says that the product of the two numbers $x$x and $1-x$1−x must be positive. A little thought leads to the realisation that $x$x must be confined to the interval $0\le x\le1$0≤x≤1.

Any number outside that interval will cause either $x$x or $1-x$1−x (but never both) to become negative, and hence the product will become negative as well. So the natural domain is given by $0\le x\le1$0≤x≤1.

 

Range from equations

To find the range you can try one of the following strategies:

• Graph the equation using technology and observe the range
• Use our knowledge of the type of graph the equation produces to understand if there are any restrictions in the vertical direction. E.g. Find turning points for parabolas.
• Rearrange the equation to '$x=$x= ....' and similar to domain ask ourselves "Is there any $y$y value the equation cannot handle?"

 

 

When looking at parabolas, unless we have been given a restricted domain, we will find that the domain is all real $x$x, also written as the set $\left(-\infty,\infty\right)$(−∞,∞), and that the range is either $\left[\text{min value},\infty\right)$[min value,∞) for concave up parabolas, or $\left(-\infty,\text{max value}\right]$(−∞,max value] for concave down parabolas. This is illustrated in the following image.  

Practice questions

Question 3

Question 4

question 5

Question 6